[EM] Pairwise Median Rating

Tue Jan 16 07:20:58 PST 2024

Ted,

>   3. Otherwise, drop ballots that don't contain ranks above last for
>        any member of the Smith Set.

Why not simply drop all ballots that make no distinction among members 
of the Smith set?

> I believe it passes LNHelp.

Douglas Woodall showed some time ago that Condorcet and LNHelp are 
incompatible.  I can't find
his proof, but it says so here:

https://en.wikipedia.org/wiki/Later-no-help_criterion

> TheCondorcet criterion 
> <https://en.wikipedia.org/wiki/Condorcet_criterion>is incompatible 
> with later-no-help.

 From your post again:
> It probably fails Participation ..

It has been known (for a longer time) that Condorcet and Participation 
are incompatible.

So since we know for sure that your method meets Condorcet, we also know 
that it doesn't meet
Later-no-Help or Participation.

Using a multi-slot ratings ballot for a Condorcet method of similar 
complexity I like:

*Eliminate  all candidates not in the Smith set.

Interpret each ballot as giving  approval to those remaining candidates 
they rate above average (mean
of the ratings given to Smith-set members).

Now, using these approvals, elect the Margins-Sorted Approval winner.*

> Has Smith//Median Rating been proposed before?
Not that I know of.

Chris Benham

>
> *Ted Stern*dodecatheon at gmail.com 
> <mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
> /Tue Jan 2 15:12:26 PST 2024/
>
>   * Previous message (by thread):[EM] [Game Theory] Iterated
>     Prisoners' Dilemma as a voting method metric
>     <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005215.html>
>   * Next message (by thread):[EM] Pairwise Median Rating
>     <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005218.html>
>   * *Messages sorted by:*[ date ]
>     <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/date.html#5216>[
>     thread ]
>     <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/thread.html#5216>[
>     subject ]
>     <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/subject.html#5216>[
>     author ]
>     <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/author.html#5216>
>
> ------------------------------------------------------------------------
> Continuing my search for a summable voting method that discourages burial
> and defection, I've come across this hybrid of Condorcet and median ratings
> that acts like Smith/Approval with an automatic approval cutoff. I'm
> calling it Pairwise Median Rating (PMR), but it could also be described as
> Smith//MR//Pairwise//MRScore:
>
>     1. Equal Ranking and ranking gap allowed (essentially a ratings method
>     with rank inferred). For purposes of this discussion, assume 6 slots (5
>     ranks above rejection).
>     2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
>     means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
>     of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
>     3. [Smith]
>        1. Compute the pairwise preference array
>        2. The winner is the candidate who defeats each other candidate
>        pairwise.
>        3. Otherwise, drop ballots that don't contain ranks above last for
>        any member of the Smith Set.
>     4. [Median Rating]
>        1. Set the MR threshold to top rank.
>        2. While no Smith candidate has a majority of undropped ballots at or
>        above the threshold, set the threshold to the next lower rank,
> until there
>        is no lower rank.
>        3. The winner is the single candidate that has a majority of
>        undropped ballots at or above the threshold.
>     5. [Pairwise]
>     1. Otherwise, if more than one candidate passes the threshold, look for
>        a pairwise beats-all candidate among candidates meeting the MR threshold.
>        (i.e. Condorcet on just the MR threshold set).
>        2.  If there is one, you have a winner.
>     6. [MR Score]
>     1. Otherwise, the winner is the Smith set candidate with the largest
>        number of ballots at or above the Median Rating threshold (their MRscore).
>
> This method is essentially Smith//Approval(explicit) with the approval
> cutoff automatically inferred via median ratings
>
> Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
> considered optional, but by doing that, you ensure Immunity from Irrelevant
> Ballots (IIB), aka the zero ballot problem that affects other Median Rating
> / Majority Judgment methods. In other words, the majority threshold is
> unaffected by ballots that do not rank a viable candidate. It is possible
> to do this summably if need be.
>
> PMR either passes the Chicken Dilemma criterion without adjustment, or
> there is a downranking strategy for defending against defection.
>
> Consider the following examples from Chris Benham's post re MinLV(erw)
> Sorted Margins (
> http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
> ):
>
> >/* 46 A>B /*>* 44 B>C (sincere is B or B>A)
> *>* 05 C>A
> *>* 05 C>B
> *>>* A>B 51-49,    B>C  90-10,    C>A 54-46.
> *
>
> With sincere ballots, A is the Condorcet Winner (CW).  With B's defection,
> there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
> threshold is 2nd place, and A and B both pass the threshold. A defeats B,
> so A is the winner and B's defection/burial fails.
>
> >/* 25 A>B /*>* 26 B>C
> *>* 23 C>A
> *>* 26 C
> *>>* C>A  75-25,    A>B  48-26,   B>C  51-49*
>
> C wins with PMR (MR threshold is first place). B would win with most
> other Condorcet methods.
>
> >/* 35 A /*>* 10 A=B
> *>* 30 B>C  (sincere B > A)
> *>* 25 C
> *>>* C>A  55-45,     A>B  35-30 (10A=B not counted),   B>C 40-25.
>
> *A wins with sincere voting. When B defects to try to win, which it
> would do with most other Condorcet methods, B wins. With PMR, C wins,
> an undesirable outcome for B.
>
> Here is another example from Rob LeGrand
> (https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
> good example for chicken dilemma resistance, but it does demonstrate
> differences from Schulze, MMPO, RP and Bucklin:
>
> # example from method description page
>   98:Abby>Cora>Erin>Dave>Brad
>   64:Brad>Abby>Erin>Cora>Dave
>   12:Brad>Abby>Erin>Dave>Cora
>   98:Brad>Erin>Abby>Cora>Dave
>   13:Brad>Erin>Abby>Dave>Cora
> 125:Brad>Erin>Dave>Abby>Cora
> 124:Cora>Abby>Erin>Dave>Brad
>   76:Cora>Erin>Abby>Dave>Brad
>   21:Dave>Abby>Brad>Erin>Cora
>   30:Dave>Brad>Abby>Erin>Cora
>   98:Dave>Brad>Erin>Cora>Abby
> 139:Dave>Cora>Abby>Brad>Erin
>   23:Dave>Cora>Brad>Abby>Erin
>
> The pairwise matrix:
>
> against
> Abby Brad Cora Dave Erin
> for Abby  458 461 485 511
> Brad 463  461 312 623
> Cora 460 460  460 460
> Dave 436 609 461  311
> Erin 410 298 461 610
>
> There is no Condorcet winner.  The Smith set is {Abby, Brad, Dave, Erin}.
>
> Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
> wins with Bucklin.
>
> In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
> the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
> score of 484 is only slightly over the 50% mark of 460.5, so the Dave
> voters hold the balance of power. Dave defeats Brad pairwise, so Dave
> voters might not be as happy with a Brad victory, and Abby might be
> able to persuade Dave voters to downrank Brad but not Abby. If
> successful, Brad drops 44 points in MRScore and is no longer in the MR
> threshold set. Abby defeats Erin, so Abby wins.
>
> * 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
>   30:Dave>Brad>Abby>Erin>Cora
>   98:Dave>Brad>Erin>Cora>Abby
> 139:Dave>Cora>Abby>Brad>Erin
> * 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
>
> PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
> I believe it passes LNHelp. It probably fails Participation and IIA.
> There are probably weird examples where changing one vote changes the
> MR threshold. But overall, I think it has a good balance of incentive
> to deter burial and deliberate cycles.
>
> Has Smith//Median Rating been proposed before? It seems like a simple
> modification to MR on its own.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20240117/5198b3a8/attachment.htm>